Final answer:
To find the number of items sold when revenue is maximized, use the vertex formula. The number of items sold when revenue is maximized is 12. To find the maximum revenue, substitute the x-value into the revenue function. The maximum revenue is $1008. To find the number of items sold when profit is maximized, use the profit function. The number of items sold when profit is maximized is 7. The maximum profit is $7. To find the break-even quantities, set the revenue equal to the cost and solve for x. The break-even quantities are 4 and 10.
Step-by-step explanation:
To find the number of items sold when revenue is maximized, we need to find the x-value that corresponds to the maximum value of the revenue function. This can be done by finding the vertex of the parabolic revenue function. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic function. Plugging in the values from the revenue function, we get x = -168/(2*-7) = 12.
Therefore, the number of items sold when revenue is maximized is 12.
To find the maximum revenue, we substitute the x-value from part (a) into the revenue function. Plugging in x = 12, we get R(12) = -7(12)^2 + 168(12) = 1008.
So, the maximum revenue is $1008.
To find the number of items sold when profit is maximized, we need to calculate the profit function and find its maximum value. The profit function is given by P(x) = R(x) - C(x), where R(x) is the revenue function and C(x) is the cost function. Substituting the given functions, we get P(x) = (-7x^2 + 168x) - (70x + 280) = -7x^2 + 98x - 280. The x-coordinate of the vertex of the profit function represents the number of items sold when profit is maximized. Using the formula x = -b/2a, where a = -7 and b = 98, we get x = -98/(2*-7) = 7. Therefore, the number of items sold when profit is maximized is 7.
To find the maximum profit, we substitute the x-value from part (c) into the profit function. Plugging in x = 7, we get P(7) = -7(7)^2 + 98(7) - 280 = 7.
So, the maximum profit is $7.
To find the break-even quantity/quantities, we need to determine the x-values for which the revenue equals the cost. This can be done by setting the revenue function equal to the cost function and solving for x. Plugging in the given functions, we get -7x^2 + 168x = 70x + 280. Simplifying this equation, we get -7x^2 + 98x - 280 = 0. Solving this quadratic equation using factoring or the quadratic formula, we find the x-values of 4 and 10.
Therefore, the break-even quantities are 4 and 10.